Clifford's theorem

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This fact is related to: linear representation theory
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Statement

Verbal statement

The restriction of any irreducible complex character of a group, to a normal subgroup, is a multiple of the sum of all conjugates in the whole group of some irreducible character of the normal subgroup.

Statement with symbols, using character-theoretic language

Let G be a finite group and N a normal subgroup of G. Let \chi be a complex irreducible character of G and \mu of N such that:

\langle \operatorname{Res}(\chi)_N^G, \mu \rangle \ne 0

Then:

\operatorname{Res}(\chi)_N^G = a \left(\sum_{i=1}^t \mu^{(g_i)}\right)

where g_i \in G and \mu^{(g)} denotes the character:

n \mapsto \mu(gng^{-1})

Further a and t are positive integers dividing the index [G:N]. In fact, t is the index of the subgroup I_G(\mu), defined as:

\left \{ g \in G| \mu^{(g)} = \mu \right \}

I_G(\mu) is termed the inertial subgroup.

Further, e divides the index [I_G(\mu):N].

Statement with symbols, using module-theoretic language

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Particular cases

Conjugacy-closed subgroups

A conjugacy-closed subgroup is a subgroup such that any two elements of the subgroup conjugate in the whole group, are also conjugate in the subgroup. If N is conjugacy-closed, then I_G(\mu) = G for any \mu and thus, in this case, the restriction of the irreducible character from G to N is simply a multiple of \mu.

References

Textbook references